I screen sift my woodstove and BBQ smoker ash and reclaim the small chunks of charcoal, that can be burned as fuel again.
If not, no biggie, charcoal can be beneficial to compost, garden and lawn.
Mostly the ash I scoop goes directly into the compost pile and gets turned in.
I try to mix it in most times, but sometimes just put on top of the pile and the next rain, or snow, will melt it right down into the pile.
Wood ash will change Ph to alkaline very quickly, but is not long lasting.
My area, Mid Atlantic, tends to run on the acidic side, so the ash can be used anywhere except near acid loving plants.
Ash can change the Ph of your compost, but again, it will eventually wash out.
If there is a drought it will add to the dryness of the compost pile, and stall the composting action due to dehydration.
If in doubt, follow the ash with a bucket or two of water and turn the pile. I'm talking about larger open air piles.
If using a barrel or bin type of composter, maybe add the ash in "doses" , mix it in, with an equivilent amount of water. I gallon bucket of ash into a 30 or 50 gallon composter, mix it in, then toss in a gallon or so of H2o to offset the dryness of the ash.